Gauss Error Function Integral
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that occurs in probability, statistics, and partial differential equations describing diffusion. It is defined as:[1][2] erf ( x ) = 1 π ∫ − x x e − t 2 d t = integral of error function 2 π ∫ 0 x e − t 2 d t . {\displaystyle erf function calculator {\begin − 6\operatorname − 5 (x)&={\frac − 4{\sqrt {\pi }}}\int _{-x}^ − 3e^{-t^ − 2}\,\mathrm − 1 t\\&={\frac − 0{\sqrt {\pi
Error Function Table
}}}\int _ 9^ 8e^{-t^ 7}\,\mathrm 6 t.\end 5}} The complementary error function, denoted erfc, is defined as erfc ( x ) = 1 − erf (
Erf(1)
x ) = 2 π ∫ x ∞ e − t 2 d t = e − x 2 erfcx ( x ) , {\displaystyle {\begin 2\operatorname 1 (x)&=1-\operatorname 0 (x)\\&={\frac Φ 9{\sqrt {\pi }}}\int _ Φ 8^{\infty }e^{-t^ Φ 7}\,\mathrm Φ 6 t\\&=e^{-x^ Φ 5}\operatorname Φ 4 (x),\end Φ 3}} which also defines erfcx, the scaled complementary error function[3] (which can be used erf(inf) instead of erfc to avoid arithmetic underflow[3][4]). Another form of erfc ( x ) {\displaystyle \operatorname 2 (x)} for non-negative x {\displaystyle x} is known as Craig's formula:[5] erfc ( x | x ≥ 0 ) = 2 π ∫ 0 π / 2 exp ( − x 2 sin 2 θ ) d θ . {\displaystyle \operatorname 0 (x|x\geq 0)={\frac Φ 9{\pi }}\int _ Φ 8^{\pi /2}\exp \left(-{\frac Φ 7}{\sin ^ Φ 6\theta }}\right)d\theta \,.} The imaginary error function, denoted erfi, is defined as erfi ( x ) = − i erf ( i x ) = 2 π ∫ 0 x e t 2 d t = 2 π e x 2 D ( x ) , {\displaystyle {\begin Φ 0\operatorname − 9 (x)&=-i\operatorname − 8 (ix)\\&={\frac − 7{\sqrt {\pi }}}\int _ − 6^ − 5e^ − 4}\,\mathrm − 3 t\\&={\frac − 2{\sqrt {\pi }}}e^ − 1}D(x),\end − 0}} where D(x) is the Dawson function (which can be used instead of erfi to avoid arithmetic overflow[3]). Despite the name "imaginary error function", erfi ( x ) {\displaystyle \operatorname 8 (x)} is real when x is real. When the error fun
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Inverse Error Function
more about hiring developers or posting ads with us Mathematics Questions Tags Users Badges Unanswered error function matlab Ask Question _ Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related error function python fields. Join them; it only takes a minute: Sign up Here's how it works: Anybody can ask a question Anybody can answer The best answers are voted up and rise to the top Why is the error https://en.wikipedia.org/wiki/Error_function function defined as it is? up vote 35 down vote favorite 6 $\newcommand{\erf}{\operatorname{erf}}$ This may be a very naïve question, but here goes. The error function $\erf$ is defined by $$\erf(x) = \frac{2}{\sqrt{\pi}} \int_0^x e^{-t^2}dt.$$ Of course, it is closely related to the normal cdf $$\Phi(x) = P(N < x) = \frac{1}{\sqrt{2\pi}} \int_{-\infty}^x e^{-t^2/2}dt$$ (where $N \sim N(0,1)$ is a standard normal) by the expression $\erf(x) = 2\Phi(x \sqrt{2})-1$. My question is: Why is it natural http://math.stackexchange.com/questions/37889/why-is-the-error-function-defined-as-it-is or useful to define $\erf$ normalized in this way? I may be biased: as a probabilist, I think much more naturally in terms of $\Phi$. However, anytime I want to compute something, I find that my calculator or math library only provides $\erf$, and I have to go check a textbook or Wikipedia to remember where all the $1$s and $2$s go. Being charitable, I have to assume that $\erf$ was invented for some reason other than to cause me annoyance, so I would like to know what it is. If nothing else, it might help me remember the definition. Wikipedia says: The standard normal cdf is used more often in probability and statistics, and the error function is used more often in other branches of mathematics. So perhaps a practitioner of one of these mysterious "other branches of mathematics" would care to enlighten me. The most reasonable expression I've found is that $$P(|N| < x) = \erf(x/\sqrt{2}).$$ This at least gets rid of all but one of the apparently spurious constants, but still has a peculiar $\sqrt{2}$ floating around. probability statistics special-functions normal-distribution share|cite|improve this question asked May 8 '11 at 20:19 Nate Eldredge 49.2k356129 I had assumed it was because you can expand both $\erf(x)$ and $\erf^{-1}(x)$ in a Taylor series about $0$, while you can't with $\Phi^{-1}$. I'm no
Du siehst YouTube auf Deutsch. Du kannst diese Einstellung unten ändern. Learn more You're viewing YouTube in German. You can change this preference below. https://www.youtube.com/watch?v=CcFUQhorgdc Schließen Ja, ich möchte sie behalten Rückgängig machen Schließen Dieses Video ist nicht verfügbar. WiedergabelisteWarteschlangeWiedergabelisteWarteschlange Alle entfernenBeenden Wird geladen... Wiedergabeliste Warteschlange __count__/__total__ The Error Function ei pi AbonnierenAbonniertAbo beenden233233 Wird geladen... Wird geladen... Wird verarbeitet... Hinzufügen Möchtest du dieses Video später noch einmal ansehen? Wenn du bei YouTube angemeldet error function bist, kannst du dieses Video zu einer Playlist hinzufügen. Anmelden Teilen Mehr Melden Möchtest du dieses Video melden? Melde dich an, um unangemessene Inhalte zu melden. Anmelden Transkript Statistik 16.913 Aufrufe 45 Dieses Video gefällt dir? Melde dich bei YouTube an, damit dein Feedback gezählt wird. Anmelden 46 6 Dieses gauss error function Video gefällt dir nicht? Melde dich bei YouTube an, damit dein Feedback gezählt wird. Anmelden 7 Wird geladen... Wird geladen... Transkript Das interaktive Transkript konnte nicht geladen werden. Wird geladen... Wird geladen... Die Bewertungsfunktion ist nach Ausleihen des Videos verfügbar. Diese Funktion ist zurzeit nicht verfügbar. Bitte versuche es später erneut. Veröffentlicht am 08.11.2013This is a special function related to the Gaussian. In this video I derive it. Kategorie Bildung Lizenz Standard-YouTube-Lizenz Mehr anzeigen Weniger anzeigen Wird geladen... Autoplay Wenn Autoplay aktiviert ist, wird die Wiedergabe automatisch mit einem der aktuellen Videovorschläge fortgesetzt. Nächstes Video Error Function and Complimentary Error Function - Dauer: 5:01 StudyYaar.com 11.854 Aufrufe 5:01 Evaluating the Error Function - Dauer: 6:36 lesnyk255 1.783 Aufrufe 6:36 Integral of exp(-x^2) | MIT 18.02SC Multivariable Calculus, Fall 2010 - Dauer: 9:34 MIT OpenCourseWare 204.132 Aufrufe 9:34 erf(x) function - Dauer: 9:59 Calculus Society -ROCKS!! 946
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